Lower bounds for the greatest possible number of colors in
interval edge colorings of bipartite cylinders and bipartite tori
Petros A. Petrosyan
Gagik H. Karapetyan
Institute for Informatics and Automation
Problems of NAS of RA,
Department of Informatics and Applied
Mathematics, YSU,
Department of Informatics and Applied
Mathematics, YSU,
Yerevan, Armenia
e-mail: [email protected],
[email protected]
Yerevan, Armenia
e-mail: [email protected]
For t ≥ 1 let N t denote the set of graphs which have an
ABSTRACT
An interval edge t − coloring of a graph G is a proper
edge coloring of G with colors 1, 2, … , t such that at least
one edge of G is colored by color i , i = 1, 2, … , t , and the
edges incident with each vertex v ∈ V (G ) are colored by
d G (v ) consecutive colors, where d G (v ) is the degree of the
vertex v in G . In this paper interval edge colorings of
bipartite cylinders and bipartite tori are investigated.
interval edge t − coloring, and assume: N ≡ ∪ Nt . For a
t≥1
graph G ∈ N the least and the greatest values of t , for
which G ∈ N t , are denoted by
w(G ) and W (G ) ,
respectively.
The problem of deciding whether or not a bipartite graph
belongs to N was shown in [2] to be NP − complete [3,4].
It was proved in [5] that if
G = C ( m,2 n )
or
G = T ( 2 m,2 n ) then G ∈ N and w(G ) = ∆ (G ) .
Keywords
Theorem 1 [6]. If G is a bipartite graph and G ∈ N then
Interval edge coloring, proper edge coloring, bipartite graph.
W (G ) ≤ d (G ) ( ∆ (G )−1) + 1 .
1. INTRODUCTION
Theorem 2 [7]. Let G be a regular graph.
1. G ∈ N iff χ ′(G ) = ∆ (G ) .
All graphs considered in this paper are finite, undirected
and have no loops or multiple edges. Let V (G ) and
E (G ) denote the sets of vertices and edges of a graph G ,
respectively. The degree of a vertex v ∈ V (G ) is denoted by
d G (v ) , the maximum degree of a vertex of G - by ∆ (G ) ,
2. If G ∈ N and ∆ (G ) ≤ t ≤ W (G ) then G ∈ N t .
In this paper interval edge colorings of bipartite cylinders
and bipartite tori are investigated. The terms and concepts that
we do not define can be found in [8-10].
(
the chromatic index of G - by χ ′(G ) , and the diameter of
2. LOWER BOUNDS FOR W C ( m,2n )
G - by d (G ) . Given two graphs G1 = (V1 , E1 ) and
AND W T ( 2m,2n ) .
G = (V , E ) with vertex set V = V1 × V2 and the edge set
Theorem 3. If G = C ( m,2 n ) then W (G ) ≥ 3m + n − 2 .
Proof. Let
G2 = (V2 , E2 ) , the Cartesian product G1 × G2 is a graph
{
E = ( ( u1 ,u2 ),( v1 ,v2 ) ) either u1 =v1 and ( u2 ,v2 )∈E2 or u2 =v2 and
( u1 ,v1 )∈E1} . The bipartite cylinder C ( m,2n )
product Pm × C 2 n ( m∈N , n≥ 2 ) and the

m
  2n
E (G ) =  ∪ E i ( G )  ∪  ∪ E j (G )  ,
i
j
1
1
=
=

 

bipartite torus
If α is a proper edge coloring of the graph G then α (e)
denotes the color of an edge e ∈ E (G ) in the coloring α .
For a proper edge coloring α of a graph G and for any
v ∈ V (G ) we denote by S (v, α ) the set of colors of edges
incident with v .
An interval [1] edge t − coloring of a graph G is a
proper edge coloring of G with colors 1, 2, … , t such that at
least one edge of G is colored by color i , i = 1, 2, … , t ,
and the edges incident with each vertex v ∈ V (G )
colored by d G (v ) consecutive colors.
V (G ) = { x (j i ) 1≤i ≤m,1≤ j ≤ 2 n} ,
is the Cartesian
T ( 2 m,2 n ) is the Cartesian product C2 m × C2 n ( m≥ 2, n≥ 2 ) .
are
)
(
)
where
E (G ) =
i
{( x
(i)
j
} {( x
, x (j i+)1 ) 1≤ j ≤ 2 n −1 ∪
E j (G ) =
{( x
(i )
j
,x
( i +1)
j
(i )
1
}
, x2( in) ) ,
) 1≤i≤m−1} .
Define an edge coloring α of the graph G in the
following way:
1. for i = 1, 2, … , m, j = 1, 2, … , n + 1
(
)
α ( x(j i ) , x(j +i )1 ) = 3i + j − 3 ;
2. for i = 1, 2, … , m, j = n + 2, … , 2 n − 1
(
)
α ( x(j i ) , x(j+i )1 ) = 3i − j + 2 n − 1 ;
3. for i = 1, 2, … , m
(
)
α ( x1( i ) , x2( in) ) = 3i − 1 ;
4. for i = 1, 2, … , m − 1, j = 2, 3, … , n + 1
(
α ( x ,x
(i )
j
( i +1)
j
) ) = 3i + j − 2 ;
5. for i = 1, 2, … , m − 1, j = n + 2, … , 2 n
(
)
α ( x (j i ) , x (j i+1) ) = 3i − j + 2 n + 1 ;
6. for i = 1, 2, … , m − 1
(
Let us show that α is an interval edge ( 3m + n − 2 ) −
2 ≤ ∆ ( C ( m,2 n ) ) ≤ 4 and d ( C ( m,2 n ) ) = m + n − 1 then
(
Theorem 4. If G = T ( 2 m,2 n ) then
{

 2m
  2n
E (G ) =  ∪ E i (G )  ∪  ∪ E j (G )  , where
 i=1
  j =1

{( x , x ) 1≤ j≤2n−1} ∪ {( x
E (G ) = {( x , x ) 1≤i ≤ 2 m −1} ∪ {( x
E (G ) =
i
α (ei ) = i .
}
Fi = α ( ( x (ji ) , x (ji+)1 ) ) 1≤ j ≤n +1 .
Clearly,
Fi = {3i −2,3i −1,…,3i + n −2} , Fi = n + 1, i = 1, 2, … , m .
It is not hard to check that
m
∪ Fi = {1,2,…,3m+ n−2} ,
i =1
(i )
j
ei ∈ E (G ) such that α (ei ) = i .
Now, let us show that the edges that are incident to a
vertex v ∈ V (G ) are colored by d G (v ) consecutive colors.
Let x j ∈ V (G ) , where 1 ≤ i ≤ m,1 ≤ j ≤ 2 n .
(i)
Case 1. i = 1, j = 1, 2 .
It is not hard to see that
S ( x ,α ) = S ( x
(i )
2 n+3− k
,α ) = {k −1,k ,k +1} ,
where k = 3, … , n + 1.
Case 3. i = m, j = 1, 2 .
It is not hard to see that
S ( x (j i ) ,α ) = {3i −3,3i − 2,3i −1} .
Case 4. i = m, j = 3, … , 2 n .
It is not hard to see that
S ( xk( i ) ,α ) = S ( x2( in)+3−k ,α ) = {3i + k −5,3i + k −4,3i + k −3} ,
where k = 3, … , n + 1.
Case 5. i = 2, 3, … , m − 1, j = 1, 2 .
It is not hard to see that
S ( x (j i ) ,α ) = {3i −3,3i − 2,3i −1,3i} .
Case 6. i = 2, 3, … , m − 1, j = 3, … , 2 n .
It is not hard to see that
S ( x ,α ) = S ( x
(i )
2 n+3− k
(i )
1
( i +1)
j
(1)
j
,α ) = {3i −k + 2 n−2,3i − k + 2n−1,
3i −k + 2n,3i −k + 2n+1} , where k = n + 2, … , 2 n.
}
, x2( in) ) ,
}
, x (j 2 m ) ) .
(
)
(
)
β ( x (j i ) , x(j+i )1 ) = β ( x (j 2 m+1−i ) , x(j +21m+1−i ) ) = i + 3 j − 3 ;
2. for i = 1, 2, … , m, j = n + 2, … , 2 n − 1
(
)
(
)
β ( x (j i ) , x(j+i )1 ) = β ( x(j 2 m+1−i ) , x(j+21m+1−i ) ) = i − 3 j + 6 n + 3 ;
3. for i = 1, 2, … , m
(
)
(
)
β ( x1( i ) , x2( in) ) = β ( x1( 2 m+1−i ) , x2( 2nm+1−i ) ) = i + 3 ;
4. for i = 1, 2, … , m, j = 2, 3, … , n + 1
(
)
(
)
β ( x (j i ) , x(j i+1) ) = β ( x(j 2 m−i ) , x(j 2 m+1−i ) ) = i + 3 j − 4 ;
5. for i = 1, 2, … , m, j = n + 2, … , 2 n
(
)
(
)
β ( x (j i ) , x(j i+1) ) = β ( x(j 2 m−i ) , x(j 2 m+1−i ) ) = i − 3 j + 6 n + 5 ;
6. for i = 1, 2, … , m
S ( x (j i ) ,α ) = {3i −2,3i −1,3i} .
(i)
k
(i )
j +1
Define an edge coloring β of the graph G in the
following way:
1. for i = 1, 2, … , m, j = 1, 2, … , n + 1
and, therefore for i , i = 1, 2, … , 3m + n − 2 there is an edge
(i)
k
(i)
j
j
For i = 1, 2, … , m we define a set Fi in the following
}
Proof. Let m ≤ n and V (G ) = x (j i ) 1≤i ≤ 2 m,1≤ j ≤ 2 n ,
… , 3m + n − 2 there is an edge ei ∈ E (G ) such that
Case 2. i = 1, j = 3, … , 2 n .
It is not hard to see that
)
from theorem 1 we have W C ( m,2 n ) ≤ 3m + 3n − 2 .
− coloring of the graph G .
First of all let us prove that for i , i = 1, 2, …
{
is a bipartite graph with
W (G ) ≥ max {3m + n,3n + m}
)
α ( x1( i ) , x1( i+1) ) = 3i .
way:
C ( m,2 n )
Remark. Since
(
)
(
)
β ( x1( i ) , x1( i+1) ) = β ( x1( 2 m−i ) , x1( 2 m+1−i ) ) = i + 2 ;
7. for j = 3, … , n + 1
(
)
(
)
(2m)
β ( x (1)
) = β ( x2(1)n+3− j , x2( 2n+m3)− j ) = 3 j − 4 ;
j ,xj
8. β
(( x
(1)
1
)
, x1( 2 m ) ) = β
Let us show that β
(( x
(1)
2
)
, x2( 2 m ) ) = 2 .
is an interval edge
( 3n + m ) −
− coloring of the graph G .
First of all let us prove that for i , i = 1, 2, …
… , 3n + m
there is an edge
ei ∈ E (G )
such that
β (ei ) = i .
It is not hard to check that
∪ ∪ S ( x (ji ) ,β ) = {1,2,…,3n+ m} ,
m n+1
i =1 j =1
and, therefore for i , i = 1, 2, … , 3n + m there is an edge
ei ∈ E (G ) such that β (ei ) = i .
Now, let us show that the edges that are incident to a
vertex v ∈ V (G ) are colored by four consecutive colors.
Let x j ∈ V (G ) , where 1 ≤ i ≤ 2 m,1 ≤ j ≤ 2 n .
(i)
Case 1. i = 1, 2, … , 2 m, j = 1 .
It is not hard to see that
Therefore, α is an interval edge ( 3m+ n − 2 ) − coloring of the
S ( x(j k ) ,β ) = S ( x (j 2 m+1−k ) , β ) = {k ,k +1,k + 2,k +3} ,
graph G .
The proof is complete.
where k = 1, 2, … , m.
Case 2. i = 1, 2, … , 2 m, j = 2, 3, … , n + 1 .
It is not hard to see that
S ( x (j k ) ,β ) = S ( x (j 2 m+1−k ) ,β ) = {k +3 j −6,k +3 j −5,
k +3 j −4,k +3 j −3} , where k = 1, 2, … , m.
Case 3. i = 1, 2, … , 2 m, j = n + 2, … , 2 n − 1 .
It is not hard to see that
S ( x (j k ) ,β ) = S ( x (j 2 m+1−k ) ,β ) = {k −3 j + 6n+3,k −3 j + 6n+ 4,
k −3 j +6 n+5,k −3 j +6 n+ 6} , where k = 1, 2, … , m.
Case 4. i = 1, 2, … , 2 m, j = 2 n .
It is not hard to see that
S ( x (j k ) , β ) = S ( x (j 2 m+1−k ) ,β ) = {k +3,k + 4,k +5,k + 6} ,
where k = 1, 2, … , m.
This shows that β is an interval edge ( 3n + m ) − coloring of
the graph G .
The proof is complete.
From theorem 2 and theorem 4 we have the following
Corollary. If G = T ( 2 m,2n ) , 4 ≤ t ≤ max {3m+ n,3n+ m}
then G ∈ Nt .
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